Abstract
In this paper, we investigate the well-posedness and solution regularity of a variable-order time-fractional diffusion equation, which is often used to model the solute transport in complex porous media where the micro-structure of the porous media may changes over time. We show that the variable-order time-fractional diffusion equations have flexible abilities to eliminate the nonphysical singularity of the solutions to their constant-order analogues. We also present a finite volume approximation and provide its stability and convergence analysis in a weighted discrete norm. Furthermore, we develop an efficient parallel-in-time procedure to improve the computational efficiency of the variable-order time-fractional diffusion equations. Numerical experiments are performed to confirm the theoretical results and to demonstrate the effectiveness and efficiency of the parallel-in-time method.
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Acknowledgements
The authors would like to express their sincere thanks to the editor and referees for their very helpful comments and suggestions, which greatly improved the quality of this paper. This work was funded by the National Natural Science Foundation of China under Grants 91630207, 11471194, 11971272, the OSD/ARO MURI Grant W911NF-15-1-0562 and the National Science Foundation under Grant DMS-1620194.
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Liu, H., Cheng, A. & Wang, H. A Parareal Finite Volume Method for Variable-Order Time-Fractional Diffusion Equations. J Sci Comput 85, 19 (2020). https://doi.org/10.1007/s10915-020-01321-x
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DOI: https://doi.org/10.1007/s10915-020-01321-x